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Mirrors > Home > MPE Home > Th. List > hlcomd | Structured version Visualization version GIF version |
Description: The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
ishlg.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
hlcomd.1 | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) |
Ref | Expression |
---|---|
hlcomd | ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcomd.1 | . 2 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) | |
2 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
5 | ishlg.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | ishlg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | ishlg.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | ishlg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
9 | 2, 3, 4, 5, 6, 7, 8 | hlcomb 26397 | . 2 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐵(𝐾‘𝐶)𝐴)) |
10 | 1, 9 | mpbid 235 | 1 ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 Itvcitv 26230 hlGchlg 26394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-hlg 26395 |
This theorem is referenced by: hlcgreulem 26411 opphllem4 26544 opphllem5 26545 opphl 26548 hlpasch 26550 lnopp2hpgb 26557 colhp 26564 cgrahl1 26610 cgrahl2 26611 cgrahl 26621 cgracol 26622 dfcgra2 26624 sacgr 26625 acopy 26627 acopyeu 26628 inaghl 26639 tgasa1 26652 |
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